Evaluate the integral: I = integral dx/((1 + x) sqrt(x))
Evaluate the integral: $$ I = \int \frac{dx}{(1 + x) \sqrt{x}}. $$
Explanation of Method
Setting $u=\sqrt{x}$ (so $x=u^2$) is the standard substitution whenever $\sqrt{x}$ appears alongside a polynomial in $x$. It converts $dx$ into $2u\,du$, and the factor of $u$ in the denominator (from $\sqrt{x}$) cancels one of the $u$'s from $dx$, leaving the standard arctangent integral $\int \frac{du}{1+u^2}$.
- Substitute $u=\sqrt{x}$ and rewrite $dx$ in terms of $du$.
- Cancel the common factor of $u$ between numerator and denominator.
- Recognize the resulting integral as a standard arctangent form.
- Substitute back and add the constant of integration.
Step 1 · Use substitution
To simplify the integral, let's use the substitution:
$$ u = \sqrt{x}, \quad \text{so that} \quad x = u^2. $$
Differentiating both sides with respect to $x$, we get:
$$ dx = 2u\,du. $$
Now, substitute $x=u^2$ and $dx=2u\,du$ into the original integral:
$$ I = \int \frac{2u\,du}{(1+u^2)u}. $$
Step 2 · Simplify the integrand
Notice that the $u$ terms cancel out:
$$ I = \int \frac{2\,du}{1+u^2}. $$
Step 3 · Integrate
The integral $\displaystyle \int \frac{du}{1+u^2}$ is a standard integral and equals $\arctan(u)$. So, we have:
$$ I = 2\arctan(u) + C, $$
where $C$ is the constant of integration.
Step 4 · Substitute back in terms of $x$
Recall that $u=\sqrt{x}$, so substituting this back, we get:
$$ I = 2\arctan(\sqrt{x}) + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u=\sqrt{x},\ x=u^2$ |
| $dx$ in terms of $du$ | $2u\,du$ |
| Antiderivative in $u$ | $2\arctan(u)$ |
Additional Notes
- The domain of $I$ requires $x > 0$, since $\sqrt{x}$ appears in a denominator and must be defined and nonzero.
- The substitution $u=\sqrt{x}$ is worth remembering whenever an integrand mixes $\sqrt{x}$ with polynomial expressions in $x$ — it often produces a clean cancellation like the one here.
- The result can be checked by differentiating $2\arctan(\sqrt{x})$ with the chain rule and confirming it returns $\dfrac{1}{(1+x)\sqrt{x}}$.
Result
The solution to the integral is: